Direct answer
A worked example of cross rates shows, step by step, how to compute the exchange rate between two currencies when neither currency pair is directly quoted in the form you need. The method uses an intermediate “bridge” currency (often but not necessarily USD) and combines two given quotes to produce the cross rate.
Cross rates matter because the same market information can be expressed through different pairs, but the resulting numbers depend on conventions and costs. A correct worked example makes every assumption explicit—quote direction, whether rates are mid-prices, and how rounding is handled.
Mechanism and definition
A cross rate is the implied exchange rate between currency A and currency C derived from two other rates, typically involving an intermediate currency B.
A common setup is:
- You have A/B (how many units of B for one unit of A, or the inverse—direction matters).
- You have C/B or B/C (again, direction matters).
- You compute A/C (the cross rate you want).
Key assumption to state in any worked example:
- Quotes are expressed with a consistent direction (e.g., “X units of quote currency per 1 unit of base currency”).
- You use the same price type for both inputs (e.g., both are mid rates).
- You ignore transaction costs and execution frictions for the math step (then discuss them as limitations).
Worked numerical example (with explicit assumptions)
Assume the following quote conventions:
- A/B means: 1 unit of currency A equals ? units of currency B.
- You want A/C.
- Intermediate currency is B.
Let currency A = EUR, intermediate currency B = USD, and currency C = GBP. Assumptions for the example:
- Use mid-style conceptual rates (no spread).
- Quote direction is exactly “base per 1 base unit”:
- EUR/USD = 1.2000 means 1 EUR = 1.2000 USD.
- GBP/USD = 1.3500 means 1 GBP = 1.3500 USD.
- Ignore fees, slippage, and rounding until the final step.
Goal: compute EUR/GBP, meaning “1 EUR = ? GBP.”
Step 1: Convert EUR to USD.
- From EUR/USD: 1 EUR = 1.2000 USD.
Step 2: Convert USD to GBP using the inverse of GBP/USD.
- From GBP/USD: 1 GBP = 1.3500 USD.
- Therefore 1 USD = (1/1.3500) GBP = 0.740740… GBP.
Step 3: Combine the conversions.
- 1 EUR = 1.2000 USD × 0.740740… GBP/USD
- 1 EUR = 0.888888… GBP.
So the worked cross rate is:
- EUR/GBP ≈ 0.8889 GBP per EUR (if you round to 4 decimals).
Limitations and failure modes (what can go wrong)
- Quote direction errors: If you accidentally treat GBP/USD as USD/GBP (inverting twice or not at all), the result will be wrong.
- Provider differences: Real quotes include bid/ask spreads and possible rounding rules. A cross rate computed from two separate quotes may not match the provider’s displayed cross rate.
- Cost and execution effects: Even if the math is correct, buying and selling at bid/ask can change the effective rate compared to the mid-based example.
- Inconsistent inputs: Using a mid rate for one leg and a traded rate for the other leg can break the assumptions and produce a cross rate that cannot be verified.
- Historical relationships don’t imply future behavior: Cross-rate relationships are a pricing identity under consistent quotes, but market conditions and costs change how you realize those rates.
Verification and next question
To verify a worked cross-rate computation independently, recompute it using the same stated assumptions:
- Start with the two input quotes.
- Apply consistent direction (base/quote meaning).
- Convert via the intermediate currency using multiplication and inversion as needed.
- Check consistency: if you compute EUR→GBP and then GBP→EUR using inverses under the same assumptions, you should return to the original conversion.
If you share your exact input quote formats (for example, whether they are base/quote or quote/base), you can test whether your own cross-rate calculation follows the same mechanics.